MinT - Best Known (136−85, 136, s)-Nets in Base 9

Best Known (136−85, 136, s)-Nets in Base 9

(136−85, 136, 81)-Net over F₉ — Constructive and digital

Digital (51, 136, 81)-net over F₉, using

t-expansion [i] based on digital (32, 136, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(136−85, 136, 182)-Net over F₉ — Digital

Digital (51, 136, 182)-net over F₉, using

t-expansion [i] based on digital (50, 136, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(136−85, 136, 2383)-Net in Base 9 — Upper bound on s

There is no (51, 136, 2384)-net in base 9, because

1 times m-reduction [i] would yield (51, 135, 2384)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 667 292555 710965 437851 040344 372596 182063 606799 689625 722994 204231 073111 224449 837275 997852 241810 630001 547459 041757 692741 119045 442305 > 9¹³⁵ [i]